Generalization of the Grothendieck's theorem

In this paper, we have obtained a generalization of the Grothendieck's theorem for the space of continuous mappings \(C_{\lambda,\mu}(X,Y)\) where \(Y\) is a complete uniform space with the uniformity \(\mu\) endowed with the topology of uniform convergence on the family \(\lambda\) of subsets of \(X\). A new topological game is defined - the Asanov-Velichko game, which makes it possible to single out a class of topological spaces of the Grothendieck type. The developed technique is used to generalize the Grothendieck theorem for the space of continuous mappings endowed with the set-open topology.